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A fermionic representation is given for all the quantities entering in thegenerating function approach to weighted Hurwitz numbers and topologicalrecursion. This includes: KP and 2D Toda $\tau$-functions of hypergeometrictype, which serve as generating functions for weighted single and doubleHurwitz numbers; the Baker function, which is expanded in an adapted basisobtained by applying the same dressing transformation to all vacuum basiselements; the multipair correlators and the multicurrent correlators.Multiplicative and differential recursion relations are deduced for the adaptedbases and their duals, and a Christoffel-Darboux type formula is derived forthe pair correlator. The quantum and classical spectral curves linking thistheory with the topological recursion programme are derived, as well as thegeneralized cut and join equations. The results are detailed for four specialcases: the simple single and double Hurwitz numbers, the weakly monotone case,corresponding to signed enumeration of coverings, the strongly monotone case,corresponding to Belyi curves and the simplest version of quantum weightedHurwitz numbers.

Fermionic approach to weighted Hurwitz numbers and topological recursion